First of all, I have to admit I had long since played a game of dota. Probably 2 months, or even more. Like always, I will get drawn back to Dota eventually. Call it passion, or even love with a mix of hate in between. But I'm not entirely sure myself now if I will ever return. This 'guide' shall serve as a closing, hopefully a good one, to my Dota experience if the worst comes to happen. It has been 3 long years, actively writing every now and then and I've realized certain aspects of Dota never changes. There is just so much to write on this wonderful game of dota that I've decided to break it down into several chapters. I have to caution you as a reader that this will not be a guide, but rather expressing my fascination in the beauty of this game. This may not be your cup of tea for a good read and I was told my topics of discussion are just too, strange. It that bothers you, then conventional guides might suit your liking. What you will find inside are topics which has remained untouched for the past few years. To deliver something original and unique, nothing revolutionary but good enough to keep people thinking. Rest assured most of the topics will be new to most of you. While I can't guarantee that you will learn much applicable content, I hope you get to gain a better appreciation of this game. Appreciation is the key to keeping the passion strong .

As for the title, "Dota: Fun to Think and Imagine", I find it very appropriate as it expresses perfectly the kind of written content in the article and eliminates any notion of being grand or pretentious. For that, thanks R.P.Feynman for your great source of inspiration.

The Chapters are divided into 2 parts, Mathcrafts and Theorycrafts. Most of the chapters under Theorycrafts are standalone topics, so you need not read it in order. As of currently, I only wrote 6 chapters but I do intend to add in more in the future, around 3-4 at least. Maybe in the near future, say in 1 months time.

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Chapter 1: What is the Fundamental Dota Mechanic?

As to scientists toiling tirelessly in search of the 'Theory of Everything', I found myself in the same position trying to piece together various elements of Dota to find some kind of relationship. Through some accidental discovery, I found that most of the numbers in Dota eventually points out in the same two directions. And that is when I realised I had struck gold.

3 years back, little did I knew the real significance of the finding till I posed the question to myself recently, "What is the Fundamental Dota Mechanic?". Almost everything in Dota revolves around it and it is what fundamentally created the strategy depth in Dota. Basing on the goal of Dota, that is to destroy, you will need to deal damage. For the defenders, you need a sizeable amount of hitpoints to withstand the damage before it hits zero. And therefore the fundamental dota mechanic is Damage and Hit points. For ease of understanding, Damage should always be referred as DPS (dmg per sec) and Hit points as EHP (Effective Hit points). Each case will be explained in great detail later on.

Thinking further, you will realise that disables, stuns and slows are greatly associated with DPS and EHP as well. It makes sense a carry is useless if he does 0 dps when disabled. The tricky part is, how do you quantify the effectiveness of a 1sec, a 2sec or 3sec disable. Or maybe a 50% 3 sec slow. You can't. It greatly varies depending on the situation and you will know it through experience. To quantify DPS and EHP however, is very much possible. And this will be the bulk of the discussion from here onwards.

Quote:

I. Understanding DPS and EHP

I came up with a visual representation to explain DPS and EHP model a couple of years ago. Below is a condensed version of what I had originally written (which was written some months later). Understand the concept is important.

Optimization in Dota

I. Introduction
We have all heard the common saying, "if you have high attack speed, go for damage items" or "If you already have a high hp, try increasing your armor". But what is the purpose in doing so? Essentially, that is optimizing be it your DPS (Dmg per sec) or your EHP (Effective HP) by balancing both Dmg/Ias and HP/Armor respectively . But how high exactly should I begin to raise my damage or armor? Most will simply reply with, .. it comes through experience in-game. How exactly then, should a new-comer to dota be able to judge such a situation correctly? He doesn't have the luxury to play a 100 games to develop such a keen sense.
This article will aim to explain how such optimization is achieved through a graphical approach

II. Fundamental Theory
In dota, there are two main components which you will be interested in optimizing, namely your DPS as well as EHP. For an easier understanding, we shall analyse how optimum DPS is achieve. First, we need to understand that there are 2 variables that makes up DPS, IAS(Increased Att Spd) and Dmg.

Imagine a scenario whereby a hero starts off with 100dmg and 0%IAS. Let's say the cost of 100dmg is the same as the cost of 100%ias. And if you are given a fixed amount of gold, would you get;

Certainly, Choice #3 is the optimum. A graphical explanation is given below:

The general case is that DPS is always maximized when it forms a perfect square. Henceforth, when it comes to optimizing DPS and EHP, we are always looking towards 'forming the square'.
Besides the 2 variables, we have to take note of the constraint too. In this case, gold cost. In the previous example, we are given that the cost of 100dmg and 100%ias is the same. What if, now, the gold cost is changed such that 50dmg will cost as much as 100%ias. Given the same scenario whereby the hero starts with 100dmg and 0%ias, it becomes:

Again, you have the choice of going +50 dmg or +100%Ias or split it 50/50 to +25dmg/+50%Ias. From the chart above, it can be clearly seen the choice of +100%Ias is the optimal choice. This explains the constraint concept where gold is taken as a factor. As a general rule, to construct an optimization diagram, you will need to know the cost ratio between the two variables Dmg and Ias. In this example, cost of Dmg to Ias is 50:100. This thus forms the perfect square. Then on, we extend the Dmg and Ias axis to as high as we want.
Don't understand? Here is an example.

III. Application
First off, I need to find the gold cost ratio between Dmg and Ias in a real Dota game. In a previous calculation I made it came out to be a ratio of 80dmg:100%Ias.
(Basically, the method I used to calculate this is somewhat an average whereby I take an average dmg per gold cost from MKB/Deso/Buriza and compare it to Ias of Hyper/Treads/MoM etc. It should be noted that it isn't exactly accurate but good enough to express it in this ratio form)

So now I have a square of 80dmg by 100%Ias. In this Scenario, I have a Lvl 7 Spiritbreaker with Lvl4 Empowering haste (+8%ms and +bonus dmg based on speed). His Items are 2x Bracers and Treads (Str)

He should therefore get more Ias.
Application is easy once you have an optimization model. They key information is the value of the square, in the case of DPS it is 80dmg:100%ias. Then on, all you need are relevant information pertaining to your hero such as dmg bonus, special hero abilities, Ias from agility etc to create your current 'Rectangle'. Once that is done, you will see clearly which area you need to increase, Ias or Dmg.
A similar model can be worked out for EHP too. For my model, I used 500hp:16.7armor (At 500hp, it will correspond to 0 armor, 1000hp~16.7armor, 1500hp~33.4armor, and so on). The ratio is derived from averaging hp items like HoT, Bracers, Vita booster etc versus Armor like Platemail, Chainmail, Armor etc.

In the previous DPS example, you may ask, what happens to BAT? Simply, BAT is not a variable. It does affect DPS in a way but since it is constant, it will not affect the result when comparing between Dmg and Ias.

IV. Limitations
The prime limitation of this theory is that the ratio of gold cost is not entirely accurate. Certain items are much cheaper (eg compare between MoM and Hyperstone) than another. This makes the ratio to vary widely. Other factors include preference of Dmg over Ias in certain situations like chasing, last-hitting, harassing and so on whereby Ias is not a major factor. DPS however, comes into effect towards late game hence such a model is only appropriate to be used at that stage of the game.
For EHP model, there is certainly a concern over HP. You wouldn't always want to optimize MHP, because it only helps to cover physical damage. A lot proportion of damage received early game are nukes from spells, hence making raw HP more valuable at that stage of the game. Again, it can only be seen useful towards lategame.

V. Conclusion
Regardless of its limitations, it is safe to say that a Tiny with 2x HoT is a fail Tiny. A Tiny with Assault Cuirass is, a smart Tiny.

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Basically, the gist of the article is: To optimize, you need to balance the various factors in equal proportions. In the graphical form, that means achieving the perfect square (or close to it). I believe it has already been explained how gold cost can affect the equilibrium ratio. To stress it further, imagine a demon edge now cost 240gold instead of 2400. Hyperstone on the other hand, remains the same 2100gold. Comparing between the two item, this translate to a 100dmg : 11.5 ias ratio. It becomes dead obvious that you should stack damage instead of attack speed as it is uberly cheaper. On the other hand, heroes with ias abilities like Viper's old frenzy which gives +80%ias (for a lack of better example) will be highly sought after. In fact, every Agi hero with a high agi gain will become popular as well. WhY? Because such abilites comes free. This essentially stretches the IAS side of the Optimization model such that you have a very broad rectangular base and the only way to form a perfect square is to add more, and even more damage into the system.
Such is the importance of gold constraint in affecting optimization. Fortunately, Dota is quite balanced. From what I've gathered in the past, the ratio is approximately 0.8dmg : 1.0ias (As per the Spiritbreaker example). I believed it hasn't changed much however.

II. A small sidetrack... on WoW

During my absence from Dota, I actually picked up WoW. Been playing it for 2.5 months so far and I've decided to take a break. It was too time consuming. Though I must admit that WoW was what inspired me to rewrite this little guide I had. As a paladin tank and a total newbie in the game, I had no choice but to read alot. You know how people bitch and whine at noobs... Of the many popular topics regarding tanks, I came across one on optimization. It spoke of the same things like balancing dodge/parry/bla bla and even suggests ratio where it is balance. I was like "Holy shiat, this is surprisingly similar to what I wrote in Dota 2 years back!". "But nobody focking understood me back then " . But I have to admit it had more application to WoW than it does to Dota. It was "need to know for WoW" whereas for Dota it was merely "nice to know".
I went on to realise that it was more reliable to WoW because of how structured the constraint in WoW is. Instead of gold costs, items in WoW every item has an item level which represents its constraint. The item level determines the amount of stats the item can hold. Each stat has a constant value, say +1 stamina will cost you 1 stat point. +1 block value will cost you 0.65 stat point. This is unlike Dota where you can't exactly place a value on how much +1dmg will cost. Compare Hyperstone, 2100g (+55ias) to MoM, 1900g (+100%IAS, +20%ms, +17%lifesteal +30%dmg). There are just too many variables (ias/ms/lifesteal/dmg/cooldown/duration/etc) making it extremely hard to be determined. It might be possible however through reverse engineering, by comparing similar items and eliminating one unknown variable at a time. For example, when comparing between IAS and DMG, I figured that IAS cost around 40 gold and Dmg was 50 gold by averaging several common items which gives a pure dmg or ias item. After which, I was able to figure other components such as Basher (which contains 2 components namely Dmg and a stun). Armor and HP can be worked out the same way and one by one you get to find the unknowns. But you will never come to an exact conclusion because this was never planned by Icefrog in the first place. His method of balancing is purely based on feel. WoW on the hand, was based on a system already set in stone but the method of assigning values was hidden. It was not until someone managed to figure the exact values for each stats through reverse engineering that Blizzard admitted it was true.

(Note: a hidden constraint besides gold do exist, item slots for example. This explains the increased cost from branch to circlets to Ultimate Orb)

III. Self-balancing Items

I thought it will be good to touch on this before I forget. Certain items are self balancing, for example an item can possess both HP + armor in equal proportions such that it saves alot of item space (remember you have 6 slots!). Of cos, a HP+armor item doesn't exist in reality, the closest example would be Vanguard. It has 2 components affecting EHP, which is raw hp and damage blocking. Individually, it will not make much of an impact, but when put together it synergizes well enough to optimize itself raising your EHP to tremendous level. This is just an example to explain the not-so-recent fad on Vanguard as a survivabilty item. Butterfly is another example of a self-balancing item. It has a good balance of Ias and Dmg (+60dmg and +60ias on AGI heroes). Now lets say we replace the 30% evasion into a buriza type of Crit (20% to deal 2.4xdmg). Essentially, this transform Butterfly from a 2-dps + 1-ehp component item into a 3-dps component item. The effect imo will be devastating as the effects stacks on top of each other. Visually, you can image the chart having 3-axes instead of 2. And instead of trying to form a perfect square, you are now trying to form a perfect cube since you have 3 factors now.

If you have played a hero arena custom map before, you'll realise Agi Heroes are extremely popular due to Agi tomes (since +agi gives +1dmg/+1ias). Str heroes can never outdps Agi types! In Heroes of Newerth, Swiftblade (Juggernaut) has very similar skills sets (Bladefury, Critical Strike, Omnislash). Except for one. Instead of Healing Ward, it is replaced with Counterattack which allows Swiftblade to retaliate (free attack) at 30% chance when being hit. Essentially, this replaces a EHP type of skill (Healing Ward) to a DPS type. What you have now is no longer a 3-component DPS skillset but 4! Needless to say, Swiftblade is 'broken' in terms of design. It is hard to predict how broken he is simply by judging from the skillset, but the problem is real. You will feel it when you play him.

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Chapter 2: There is no such thing as diminishing returns in Dota

So far I've covered the basic relationship of Attack Speed + Dmg into DPS. Likewise, Armor + HP for EHP. The next step will be to show how Criticals, Armor Reduction, Base Attack Time, etc relates to DPS and how miss chance, Block chance, Spell resistance etc relates to EHP. Before that however, I have to make this point - "There is no such thing as diminishing returns in Dota". This is particularly important because it helps in understanding how the various factors stack.

Ok so what is Diminishing Return? Basically diminishing return comes into play when every additional unit of a component adds less utility than the previous one. A hypothetical example would be buying 3 claws of attack. The first gives +9dmg, the 2nd only +8dmg and the 3rd, adds only +7dmg. Some people however confuse stacking of Criticals, bashes (in the past), and damage blocking as stacking diminishingly. That is only half true. If you were to compare to games like WoW or Diablo2, these games are heavy on Diminishing Returns Formulas. The difference between the two is that the Blizzard Games (D2 and WoW) relies on intentionally-engineered-formulas to create the diminishing returns effect usually with use of exponential type formulas Whereas in Dota, diminishing returns takes place as a result of a probability chance phenomenon which we often see in the real world. In other words, Icefrog did not design any formula to create the diminishing effect on stacking Critcals, Evasion etc. It was already there in the first place.

I. What is the probability chance phenomenon?
One of the most common type is the calculation of bashes, critical strikes and blocking. A common answer given will be in a form of this equation:

(Note: Bashes no longer stacks, neither normal bashes nor pseudo bashes. The example below is based on Bash as I feel it is easier to drive the point. Basing on Critical strikes or blocking involves more complexity as you need to take into account the Dmg multiplier (for Crit) and/or Block dmg (for block) aside from the chances to proc.)

What the equation doesn't explain is where the missing lost chances went to. The answer lies in the probablity problem itself. Here you need to break it down into its fundamental components. Here I will examine the 3x basher case to examine where the lost chances amounting to 6.4% went to:

1st case: %Chance of not proccing at all -- (0.85)(0.85)(0.85) = 0.614
2nd case: %Chance of proccing at only 1 bash -- (0.15)(0.85)(0.85)*3 = 0.3251
3rd case: %Chance of proccing 2 bashes at the same time -- (0.15)(0.15)(0.85)*3 = 0.05738
4th case: %Chance of proccing 3 bashes at the same time -- (0.15)(0.15)(0.15) = 0.003375

If we add all the 3 cases where it procs, you get a total of 38.6% chance. If you take (1-%chance of not proccing at all) you will still get 38.6% chance.
More importantly though, you get to see where the missing chances went to. Look at 3rd case. What happens when 2 bashes proc at the same time? Only 1 of the bash will take effect, the other will be wasted. The consequence of this wasted bash results in the lost of %chances.
In other words, for the 3rd case, you lose a 5.738% chance to bash (One useful bash, the other is wasted). For the 4th case, you lose 0.675% (0.3375*2) due to 1 useful bash and 2 wasted bashes. The total wasted chances hence adds up to 6.4%.
Therefore the fundamental cause of the diminishing returns is due to the fact that only a single instance of the effect (bash/crit) can occur if more than one effect (bash/crit) occurs at the same time.

Here is a simple example.
Assume you are Faceless void with Backtrack (25% pseudo-evasion from backtrack) with a Butterfly (30% evasion). Both stacks, 'diminishingly'. The diminished value is a result of lost chances when both Butterfly's evasion and Backtrack's evasion procs. Because, even in the real world, there is no such thing as missing TWICE in a single attack. When you attack, it is either a miss or a hit. The same concept is applied to Blocking chance (you can only block once in a single hit!), Critical (critical once in a single attack) etc. Spell resistance stack in the same manner as well, eg an Anti-mage with a Hood of Defiance. Though it is pretty strange to visualize it in the same manner. You can't say the same as "you can only spell-resist a spell once on every spell attack" because it isn't spell-evasion. Strange but that is how it is.

To sum it up, the cause of diminishing effect in Dota is the result of a natural phenomenon resulting from a probability case where chances are lost due to multiple instances proccing at the same time.

The point I want to drive across is that a lot of people are having misconceptions on what diminishing returns in Dota is. The term is flung around loosely without really much understanding it and has generated alot of misconceptions and confusion over the years. People are more concerned with getting the final results rather than understanding the process in arriving to the solution. The next chapter will deal with various forms of stacking, and getting the concept of 'diminishing returns' right will assist in understanding it better.

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Chapter 3: Types of Stacking

Quoting myself earlier:

Quote:

I've covered the basic relationship of Attack Speed + Dmg into DPS. Likewise, Armor + HP for EHP. The next step will be to show how Criticals, Armor Reduction, Base Attack Time, etc relates to DPS and how miss chance, Block chance, Spell resistance etc relates to EHP

In general, all the factors described above stacks multiplicatively with each other. Additive stacking (Addition) is simple. If you buy a Vitality booster, you get +250hp. Buy a new one, another +250hp. It adds up nicely. On the other hand, when you attempt to balance HP and Armor to optimize the EHP value, the result is a multiplicative stacking. Generally it means you gain more that you would additively. Like as if you gained hidden 'bonuses' as you draw closer to the 'perfect square'. To explain it in a simpler fashion, recall the earlier example:

Do you get
1) 100 Dmg or
2) 100% IAS or
3) 50dmg and 50% IAS

You get the same final amount for 1st and 2nd case, but larger for the 3rd case. Because the it adds 'multiplicatively'. Instead of HP + HP = Final HP as in the Vit booster case, you get DMG * IAS = Final DPS.

If we analyze the components in DPS, we have:
Dmg, IAS, Critical strike chance, Armor Reduction and Base attack Time

These components affects the final DPS but we need to have a common form in representing their values. For example, a 24% chance to crit from Buriza has no relationship to a hero with 1.55sec base attack speed. The trick is to convert them into "Dmg Increase" to represent them in a common form. Hence, buriza becomes +24% dmg and 1.55sec BAT becomes +9.67dmg increase (1.70 BAT divide by 1.55). Hence, Critical strikes, Armor Reduc, and BAT are represented in this form, called 'Multipliers'. The base components, DMG and IAS still are expressed in their original form.

Now the question is, how do you optimize this string of equation?
The same way you do using just 2 components, Dmg and Ias. Lets say the equilibrium ratio is 100dmg:100ias, then by the above example, you will need to add in +50%ias to make it 2.00. But bear in mind the constraint factor which is gold. The equilibrium ratio could in fact be close to 80dmg:100ias:1.24(crit) instead. I don't know. The only way to find out is to figure the approximate value of the crit_chance component. (Recall the reverse engineering method mentioned in the earlier chapter). For BAT however, there is no exact equibrium ratio relationship because it is an unpurchasable skill! No such items exist. Armor reduction do have a cost, but it is rather complicated so I will leave it to later chapters to discuss. Earlier, I mentioned that to optimize you need to form a square, and if it has 3 components you try to form a cube. Now it is hard to visualize how a 4-dimension system looks like, so we have to fall back on numbers. Basically to form the square or cube, you need the values to be the same. Eg, Area = 2*2 or cube = 3*3*3. Thus for the example above, with a hypothetical equilibrium ratio of 200(dmg) : 2.00(Ias) : 2.00(Crit) : 2.00(Armor_red) [BAT omitted], the same balancing rules apply.

For example, if you currently have 200dmg, 2.00Ias, 1.00Crit and 1.00Armor_red, given a 1.00 amount worth of stats to be distributed among the 4-components, you will split it evenly 50/50 to achieve 1.50Crit and 1.50Armor_red. Play around with the numbers, and you will find it is always true.

That extends the concept of optimization from "trying to form the perfect square" to "balancing each components in equal proportions". Equal proportions in this case has a specific meaning ie, the equilibrium ratio. So what about BAT. Since it has no equilibrium ratio value, how do we quantify its importance. You can't. You should however always be on the lookout of skills on heroes which grants such bonuses. Another form of multiplier not mentioned is +Dmg auras.

II. A little sidetrack... on Aura Stacking

Every once in a while, someone in the forum will come up and propose an idea on a specific Aura stacking type of strategy. For example, a team consisting of Drow (Trueshot), Luna(Lunar Blessing?), Troll (Ultimate), VS(Command Aura), etc. Does it work? Yes. But does it work well? Maybe. Obviously, the problem with such a strategy is that you are stacking too much of the same component. What you need to do is to diversify the skills into other areas as well so as to "balance each components". Right away you can spot that from the above example, 3 of them are dmg-auras while one one is an Ias type. Try taking away some of the Dmg-auras and replace it with suitable heroes which has armor reduction abilities. It will work, more effectively.

III. Stacking of same type of EHP Multipliers

Under the dimininishing returns chapter, it was found that the lost chances are present due to the probability phenomenon. This lost chances however does not in any way diminishes the final value, in fact, it multiplies. For example, an Anti Mage with Hood (30%SR), Innate Res.(25%), Spell Shield (40%) gives a total of 68.5% in Spell Dmg reduction. It doesn't stack linearly so you won't be expecting it to be 95%. However, from an EHP point of view, the effect is multiplied greatly to give a total Spell Res. Multiplier of 3.17.
There has been many views circulating around saying that stacking Hood on AM is ridiculous as the effect is diminishing, and I know very well that isn't true. Getting Hood on AM makes him REALLY beefy against nukes. Fundamentally, the SR are calculated by multiplying the multipliers of each component ie:

A great example is the use of stout shields on Axe. I was quite fascinated with the concept of Backlaning Axe carrying 2 stout shields. Back then stout shield had 40% chance to block 30dmg. Technically, this is equivalent to 40% Evasion
as lane creeps deals less than 30dmg, hence block mitigates the full dmg. Assuming a 550hp Lvl 1 Axe, the total EHP can be calculated in two ways:

And so there, you have a 1528HP hero camping behind your lane standing his ground, unstoppable. And since his HP multiplier is 2.78 (from the 2 stouts), any form of regen will be amplified by that factor as well. Hence a 400HP flask effectively heals him for 1112HP. Nice cheating.

Stout shield has since been nerfed to 60% chance to block for 20dmg. Personally written by Icefrog "the total dmg blocked hasn't been changed" since on average, both the old and new blocks for only 12dmg. It is quite misleading because if you were to stack 2x stouts, it wouldn't give the same results. The old version will block total of 19.2dmg on average while the new blocks 16.8dmg.

The above table may prove useful in getting a feel of the common Multipliers you will see in game for EHP based components (Evasion, block, Armor, etc). It is applicable for DPS components such as Critical strikes and BAT.

III. Misconception on Critical Strike and Base Attack Time (BAT)

Over the years, I have read discussions involving these terms and at times they are either expressed wrongly, or the person simply doesn't understand the mechanics behind it. A common expression in guides, "Buriza is a good item choice because it synergizes with your high damage. Bigger red numbers!". Or "Buriza is good for heroes with high attack speed because you get to crit more often". That is where the confusion sets in. Readers get the idea that only a single factor, Dmg for the first case, and Attack Speed for the second affects the final damage when in fact both factors matter (ie DPS).

Likewise for BAT, some people gets confused thinking that adding more speed will improve his overall attack speed greatly because of his already low BAT (Eg for Anti-Mage) and hence his DPS. And they find it hard to accept that adding more damage will have the same effect. They just get so confused that when they think low BAT, increasing attack speed immediately pops into mind. Or when playing Phantom Assassin, they are just so obsessed in getting more damage to get bigger red numbers. Arguably, they are doing the right thing since PA's blink strike has been buff with an IAS component providing a self-balance with the additional damage from items. But still, they have the wrong concept.

I. The Damage Reduction Table
I think this is a very interesting topic of discussion primarily because even after 5 (or more years) since Dota was first introduced, Armor Reduction effects hasn't been pretty much understood. When you thought every single thing about Dota has been discovered, you get stumped trying to explain how armor reduction effects works. First off, we already know Armor reduction falls under the DPS category therefore we need to find a way to convert it into a +%DMG increase format. Most people start off wrong as they rely solely on how Armor and EHP works (recall the +6% increase in EHP for every point of armor) trying to explain Armor reduction. You need to go a step further to find the difference prior to armor reduction and after the reduction.

Steps to finding the +%Dmg increase
1: Find the EHP multiplier of opponent's armor before reduction
2: Find the EHP multiplier of opponent's armor after reduction
3: Take the ratio of before and after (ie take before divide by after). The results is the +%Dmg Multiplier.

To be able to calculate the increase is all well and good but it gets pretty dumb if you are always punching numbers on your calculator everytime you play an Armor Reducing Hero. Like the EHP/DPS theory, the keypoint isn't on how proficient you are at doing the calculations but rather understanding the concept as a whole. Likewise, for Armor Reduction Effets, we are interested in some kind of pattern or a general rule of the thumb from which we can rely on. A way to go about it is to make a spreadsheet on all the possible values and their results. Before that however, we have to take note that Armor works differently at -ve armor when compared to +ve armor.

Basically, the difference is that at +ve Armor values increases linearly by 0.06 EHP whereas at -ve Armor values it decreases diminishingly eg. at -1 armor it reduces EHP by 0.057, 0.047 at -2, 0.041 at -3, etc. The final EHP value never reaches 0 as the dmg increase can never reach 100% (else you have 0 EHP and die instantly from the reduction effects). Even if that was possible, it wouldn't be because there is a -ve armor cap at -20 Armor.

As you would have expected, the EHP converges to zero at very large -ve Armor values. Basing from the two formulas and through the use of a spreadsheet, we get:

Also take note the the formulas refers to different meanings, one is for dmg increase, the other is dmg reduction. You have to convert into the same type, either make both in a form of Dmg Reduction, both a Dmg Increase, or both a Dmg Taken form. I chose Dmg Taken and through some manipulation results in the following formulas:

At positive values, damage taken is basically the opposite of Dmg reduction. For example a dragonknight with 11 armor will have 40% dmg reduction which is 60% Dmg taken. If Slardar casts Amplify damage for -15 armor, his new armor(-4) will result in a +21.9% increase in damage taken which is 121.9% of Damage taken. By taking the ratio of After/Before, you get 121.9/60 = 2.03 (+103% dmg increase). With this method, you can find the resulting +%Dmg increase at every value.

From the table, we can identify several patterns:

1: There is a maximum value for Dmg increase for each value of Armor reduction. We call it the 'peak' and the region around it are where the high efficiency band is (the bright green parts). As a general rule of the thumb, this peak forms when the Armor of the enemy is approximately 0.7 times that of the Armor Reduction effects. For example, a -15 armor from Slardar's Amplify Dmg has maximum effciency against heroes with 10.5 Armor(See Table).

2: As a consequence of the 0.7 times rule, the Efficiency band and peaks shifts to higher Armor values as the armor reduction effects gets higher. This will mean two things; firstly if you are facing against a high armor team (eg heroes with armor buffs like Treant and Lich), you will need huge amounts of armor reduction effects to keep in par with their high armor to achieve as close to the high efficiency band as possible. Conversely, if a team is made up of squishies (mostly Int casters and Agi heroes which stacks dps), you might end up Over-stacking armor reduction effects. When that happens, you gradually fall off the efficiency band into the red region as diminishing returns starts to take effect. You are still doing more damage than before but you are wasting your gold and resources(as in hero choices) here where they can be better spent elsewhere. For example, you can switch Desolator for Buriza which has a +Crit Multiplier which further amplifys the overall Dps. Resources wise, you shouldn't focus too much forming a line-up around Armor reduction effects. You can replace some heroes for other strategic roles. Usually, a Slardar + another Armor reduction hero is sufficient. Just don't overdo it.

3: Every -5 Armor reduction increase results in an approximately +33% increase in damage. This figure is useful to remember as it provides you with a quick way to estimate your damage output in game, or when theorycrafting.

Damage Increase represented in graph form.

II. Low Armor Paradox

Let's say you are faced against a high armor reduction team and you are a squishy tanker with low armor. You know very well that they fall under the red-band efficiency (they have huge armor reduction effects amounting up to -30 whereas you are at 10armor) and if you were to increase your armor, this will effectively raise their efficiency to the green region. It would be counter-productive to some extent. Alternatively, you can increase your HP instead since armor reduction is not affected by HP in any way. You will soon realise there will be another problem as stacking only HP will result in non-optimization in EHP due to imbalance in the Armor:HP ratio. So what do you do?
My suggestion will be to stack HP and get Blademail as a tanker. Forget raising Armor entirely. The beauty in Blademail is that what they deal is what they get. The Dmg return is pure. If you get hit for 200dmg, Blademail returns the same amount, unreduced. It completely negates Armor effects. If you are an Int squishy, the surest way to survive such a line-up is by getting Ghost Scepter.

*What you deal is what you get basically means if you lose 200hp, the enemy will lose 200hp as well. The 200hp dmg returned does not undergo dmg reduction from his armor value. Additionally, it doesn't matter if you have 50% armor reduction or 0% armor reduction. If you have 50%, you lose 100hp everytime while returning 100hp in damage. At the of the day, you will still dish out your maximum hp if you die be it with 50% armor reduction or 0%. It is only a matter of time, the former takes twice as long before it happens. More importantly however, you would want to lose HP as fast as possible in bursts to make full use of the limited 3sec duration of the blademail return effect. Additionally, the attack will have less time to respond and change targets before realizing it is too late.

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Chapter 5: Momentum Theory

Couple of years ago while I was still playing beta games, I noticed we often get into arguments over unbalanced teams. Games are most of the time lob-sided with one team practically steamrolling the other. We came up with solutions such as assigning Captains but the same problem will surface. It got me thinking trying to find a reason to explain this phenomenon. I roughly had the idea behind it but I was lacking something to explain it really well. Till one day, an idea struck me.

The model above fits perfectly in trying to explain the Momentum Theory. It should called the Momentum model or Snowball effect model, but I somewhat named it the Slippery Slope model. I can't exactly recall why, probably because I had the idea while reading an article about a Slippery Slope argument. It isn't appropriate, I know. Nomenclature aside, the model pretty much explains itselfs. You can imagine a ball on top of a hill. A push on either side will roll it in that direction. As the ball rolls down, it gains momentum and gets faster and faster. As momentum builds up, it becomes increasingly hard to resist the force as it rolls downwards. Apply this to a Dota context and you can imagine as the left side being Sentinel while Scourge is on the other. The ball starts at equilibrium on top of the hill before game starts. Ganking is one of the most critical factors in generating momentum during early. Thus, kills becomes credible indicators on how the ball will move.

For example, a Solo Mid Tinker from Sentinel managed to grab Firstblood against Shadow Friend will start pushing the ball into his team's favour (towards Scourge). Consequently, he passed by the river to grab a rune and headed up to top lane. With the aid of Allied disables, he managed to get a doublekill and the 3 of them promptly pushed down the top tower. In this series of events, Tinker capitalized on Shadow Friend's death to scout for runes. Due to sheer luck, a rune spawned and again he used that advantage to gank the Top Lane. After the doublekill, the situation opens up a great opportunity to take down the top tower and they did. This refers to Momentum. You need to keep the flow going by constantly capitalizing on opportunities that open up every now and then. Many a times, players simply forgo such opportunities. Seize it. Or the ball won't gain momentum!

Another feature of the Momentum model is the impact it has when you succesfully gank over and over again. As opponent heroes die, they lose out on gold, farming time and levels. Conversely, killing grants you more xp, gold and farming time in the lane. If you are lucky, pushing down a few towers is possible as well. If this keeps going on like a vicious cycle, the losing team gets poorer and poorer (and underleveled) while the other team gets richer and richer (and overleveled). The ball gains momentum this way because it gets increasingly easier to 'push' it in your favour. Killing underleveled heroes takes little effort. This is the dominance effect.

The model is a good representation for Early Game behaviour and it could very well predict how your team will transit into Mid-game. If your early game dominance is great with team scores going 20-2, rest assured the game will wrap itself by the 20-30th mins. It happens everywhere, from common pubs to tournament games. Differences in skills isn't the sole explanation for a team to lose badly. A good team can perform badly because they were outplayed and outpicked. They lost against a relatively stronger line-up (note: relatively). And line-ups are the consequence of this momentum theory. For example, if you are going for an early push line-up it basically means you will have a good early game. Opportunities will come by easy (eg. taking down towers, winning teamfights, early ganks) and dominating the opposing team becomes increasingly easier as well because it is quite easy to get an early streak of successful ganks thereby guranteeing your success in future ones. Gaining dominance with such a lineup is easy, the difficulty comes in maintaining the dominance.

Maintaining dominance is all about momentum. Someone in your team might just go "Oh Hey, we're winning, imma go farm". Or the usual "imma go make dagon for the lulz!" attitude. The moment you slacken off, your opponents will take the opportunity to recover. Recovery doesn't always require the other team to make mistakes. The losing team themselves can work towards recovery, needless to say with more effort and teamwork. The key is to disrupt their momentum and this can be achieved through a sudden change in playstyle, for example sticking together to get less ganked, sudden usage of wards, a sudden change from aggresive to passive play, etc. Do whatever it takes to change your playstyle. The reason you are being 'dominated' is because you are acting in a predictable manner making it too easy for your opponent to read and counter you.

I must admit however, the momentum model loses its effect and influence as game progress into the latter stage when more complex factors begin to take effect. Mistakes play a larger role in determining the game outcome. What you did wrong matters more than what you did right much to everyone's disappointment. For example, a failure on Earthshaker's part to Echo Slam can cost the game for the team. This isn't so much of a problem during early game. At most, it will only result in a loss of a tower during the defence. The dominance game is more forgiving. You can afford to make mistakes, but do note that these mistakes add up and will have a consequence on the overall outcome on the momentum theory.

And that, is what makes Dota interesting. To be able to make predictions based on how the game shifts towards a team based on how well they have been doing throughout the game, and not based on Luck factors like how the poor guy broke his "C" key during that last teamfight that prevented him from letting off his Echo Slam. And, could this model work in other strategy games as well?

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Chapter 6: Force Multiplier Effects in Dota

Force Multiplier is a term that commonly spring up in military discussions. Basically the idea is to have better technology, weaponry, training, or even taking advantage of terrain and weather to multiply the troops' fighting effectiveness.

Take a scenario where 2 platoons of soldiers are facing against each other. One has a terrain advantage (say they are defending on top of a hill) while the other is in inferior ground. Let's say the terrain multiplier is 1.5, thereby increasing the defending troops effectiveness by 50%. This is as good as having one and a half platoons against a single platoon, if counted men for men. And hence the term "force" and "multiplier". You often see this in WWII-based games, where this concept is widely used. For example, a small team with a MG-42 (better technology) can easily mow down an entire section of soldiers (larger force).

Force multiplier in Dota can be seen in several aspects of the game, but I want to draw into attention just a single component which I feel is most crucial. In dota, everyone is well-informed that a good line-up of heroes consists of a mixture of Carries, Gankers and Support. A brief description of the 3 roles:

Carries ~ Heroes with strong late-game prowess but weak early game. Highly dependable on a good farm.
Gankers ~ Heroes with strong early-game prowess but may lose effectiveness towards late-game. Most heroes of this type have low dependency on items.
Supporters ~ Heroes primarily focused in either buffing allies, serving as a warder, or possessing key spells that can make or break a game.

In practice, most heroes are hybrids, falling under more than one category. For example, PoTM can be a semi-carry (other role is ganker). Lion is a ganker-support etc.

Let's assume a pool of available heroes to choose from --

Carries:
Gankers:
Supporters:

This force multiplier concept goes like this:
If you pick the same hero type (eg a carry), the net strength caluculation will involve an addition. If you pick another type, it will have a multiplier effect. (note: For each component (carries/gankers/supporters) you have a base multiplier of 1. This means if no heroes are picked, 1*1*1 = 1.

The 1st case:
Picking all carries with no support or gankers is common in our usual pubgames. Such games will quickly evolve into insufficient farm for everyone and poor early-game control (due to zero ganking presence, you will get ganked instead). It is a weak line-up that possess zero multiplicative effects. (weak synergy between heroes, and in some cases counter-prodcutive eg too little farm.)

The 2nd case:
A more balanced lineup comprising of at least 1 hero on each type, thereby amplifying each other's effectiveness (ie synergize). For example, CM provides ample mana regeneration for her mana consuming allies. Clock and his ganking team can offer strong early-game ganking to give room for SF to farm safely.

In comparison, when pitted against each other, the 2nd team is likely to win the game given their net strength of 16 versus 6.

A more improved version of this concept would be to assign each hero to 2 or more components. For example, we can assume Shadow Fiend to be a carry and a ganker because in practice, he is a hybrid. Whereas a Spectre can assumed a full 100% carry role while Clockwerk can a ganker cum carry cum supporter, with 3 roles in 1 (if the game allows him to be).

And here we have a more defined strength measurement by basing them as hybrids.

I know the concept of synergy in a team lineup has been discussed time and again but what I have here is just an alternative, or rather, a visual explanation of how the multiplicative effects of synergy can be quantified. It is merely a concept to help you better understand the inner workings of the game, especially for players who are new to dota. I won't go into details of how an ideal lineup should look like, I don't want to turn this into a guide.

A general rule of the thumb however, is to have:
1x hard carry
3x gankers/semi-carries
1x support (for warding and/or your choice of key spells, eg Warlock for Infernal/healing, Chen for strong early push, ES/Pitlord for anti-push, Omni for Repel/GA, Tide for intiating with Ravage)

The beauty in this concept is that it is synonymous with the optimization concept we spoke of in the earlier chapters. Optimization concerns the micro details of a hero whereas the Force Multiplier concept deals with the macro aspects of the game, such as team lineup and team synergy.

Closing
I have definitely enjoyed writing this little book. I hope the read excites you in some way. I know some of the things written here are pure common sense. Like how we should get more attack speed when we have decent damage. But when you start questioning why it has to be that way, you might find something interesting. And from that something interesting you may uncover even more interesting stuffs.

There are plenty of people whom I'd like to express my thanks. The list is incredibly long and I might miss out someone here and there, so I figured it will be better to go without a list! My thanks to everyone who reads my guides, willing to open their mind to my ridiculous ideas, the 20 odd people who plays together with me in irc over the past 3 years, lz for wow, all the PMs I received on guide-related matters be it in a form of critic or encouragement, everyone whom I have cited their work in one form or another. Thanks.
It's time for me to move on to another game